<\/span><\/h2>\nQuestion 1.
State Euclid\u2019s division lemma.
Solution:
Euclid\u2019s division lemma:
Let a and b be any two positive integers, then there exist unique integers q and r such that
a = bq + r, 0 \u2264 r < b
If b\\a, then r = 0, otherwise x. satisfies the stronger inequality 0 < r < b.<\/p>\n
Question 2.
State Fundamental Theorem of Arithmetic.
Solution:
Fundamental Theorem of Arithmetics :
Every composite number can be expressed (factorized) as a product of primes and this factorization is unique except for the order in which the prime factors occur.<\/p>\n
Question 3.
Write 98 as a product of its prime factors.
Solution:
![\"1\"](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/1.png\")
<\/p>\n
Question 4.
Write the exponent of 2 in the prime factorization of 144.
Solution:
![\"2\"](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/2.png\")
<\/p>\n
Question 5.
Write the sum of the exponents of prime factors in the prime factorization of 98
Solution:
98 = 2 x 7 x 7 = 21<\/sup>\u00a0x 72<\/sup>
Sum of exponents = 1 + 2 = 3<\/p>\nQuestion 6.
If the prime factorization of a natural number n is 23<\/sup>\u00a0x 32<\/sup>\u00a0x 52<\/sup>\u00a0x 7, write the number of consecutive zeros in n.
Solution:
n = 23<\/sup>\u00a0x 32<\/sup>\u00a0x 52<\/sup>\u00a0x 7
Number of zeros will be 52<\/sup>\u00a0x 22<\/sup>\u00a0= 102<\/sup>\u00a0two zeros<\/p>\nQuestion 7.
If the product of two numbers is 1080 and their H.C.F. is 30, find their L.C.M.
Solution:
Product of two numbers = 1080
H.C.F. = 30
![\"3\"](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/3.png\")
<\/p>\n
Question 8.
Write the condition to be satisfied by q so that a rational number\u00a0pq\u00a0has a terminating decimal expansion. [C.B.S.E. 2008]
Solution:
In the rational number pq, the factorization of denominator q must be in the form of 2m<\/sup>\u00a0x 5n<\/sup>\u00a0where m and n are non-negative integers.<\/p>\nQuestion 9.
Write the condition to be satisfied by q so that a rational number\u00a0pq\u00a0has a non-terminating decimal expansion.
Solution:
In the rational number pq, the factorization of denominator q, is not in the form of 2m<\/sup>\u00a0x 5n<\/sup>\u00a0where m and n are non-negative integers.<\/p>\nQuestion 10.
Complete the missing entries in the following factor tree.
![\"4\"](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/4.png\")
Solution:
![\"5\"](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/5.png\")
<\/p>\n
Question 11.
The decimal expression of the rational number\u00a04324\u00d753\u00a0will terminate after how many places of decimals. [C.B.S.E. 2009]
Solution:
The denominator of\u00a04324\u00d753\u00a0is 24 x 53 which is in the form of 2m<\/sup>\u00a0x 5n<\/sup>\u00a0where m and n are positive integers
4324\u00d753\u00a0has terminating decimals
The decimal expansion of\u00a04324\u00d753\u00a0terminates after 4 (the highest power is 4) decimal places<\/p>\nQuestion 12.
Has the rational number 44125\u00d757\u00d772 of a terminating or a non-terminating decimal representation? [CBSE 2010]
Solution:
![\"a\"](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/a.png\")
<\/p>\n
Question 13.
Write whether\u00a02\u221a45+3\u221a202\u221a5\u00a0on simplification gives a rational or an irrational number. [CBSE 2010]
Solution:
![\"b\"](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/b.png\")
<\/p>\n
Question 14.
What is an algorithm?
Solution:
Algorithm: An algorithm is a series of well-defined slips that gives a procedure for solving a type of problem.<\/p>\n
Question 15.
What is a lemma?
Solution:
A lemma is a proven statement used for proving another statement.<\/p>\n
Question 16.
If p and q are two prime numbers, then what is their HCF?
Solution:
If p and q are two primes, then their HCF will be 1 as they have no common factor except 1.<\/p>\n
Question 17.
If p and q are two prime numbers, then what is their LCM?
Solution:
If p and q are two primes, their LCM will be their product.<\/p>\n
Question 18.
What is the total number of factors of a prime number?
Solution:
The total number of factors of a prime number is 2, first 1 and second the number itself.<\/p>\n
Question 19.
What is a composite number?
Solution:
A composite number is a number that can be factorized into more than two factors.<\/p>\n
Question 20.
What is the HCF of the smallest composite number and the smallest prime number?
Solution:
We know that 2 is the smallest prime number and 4 is the smallest composite number
HCF of 2 and 4 = 2<\/p>\n
Question 21.
HCF of two numbers is always a factor of their LCM (True \/ False).
Solution:
True.<\/p>\n
Question 22.
\u03c0 is an irrational number (True \/ False).
Solution:
True as the value of \u03c0 is neither terminating nor repeating.<\/p>\n
Question 23.
The sum of two prime numbers is always a prime number (True \/ False).
Solution:
False. The sum of two prime numbers can be a composite number
e.g. 3 and 5 are prime numbers but their sum 3 + 5 = 8 is a composite number.<\/p>\n
Question 24.
The product of any three consecutive natural numbers is divisible by 6 (True \/ False).
Solution:
True.<\/p>\n
Question 25.
Every even integer is of the form 2m, where m is an integer (True \/ False).
Solution:
True, as 2m is divisible by 2.<\/p>\n
Question 26.
Every odd integer is of the form 2m \u2013 1, where m is an integer (True \/ False).
Solution:
True, as 2m is an even number but if we subtract 1 from it, it will be an odd number.<\/p>\n
Question 27.
The product of two irrational numbers is an irrational number (True \/ False).
Solution:
False, as it is not always possible for the product of two irrational numbers be also an irrational number, it may be a rational number
for example
\u221a3 x \u221a3 = 3, \u221a7 x \u221a7 = 7<\/p>\n
Question 28.
The sum of two irrational numbers is an irrational number (True \/ False).
Solution:
False, as it is not always possible that the sum of two irrational is also an irrational number, it may be a rational number also.
For example
(2 + \u221a3) + (2 \u2013 \u221a3) = 2 + \u221a3 + 2 \u2013 \u221a3 = 4<\/p>\n
Question 29.
For what value of n, 2n<\/sup>\u00a0x 5n<\/sup>\u00a0ends in 5.
Solution:
In 2n<\/sup>\u00a0x 5n<\/sup>\u00a0,
There is no such value of n, which satisfies the given condition.<\/p>\nQuestion 30.
If a and b are relatively prime numbers, then what is their HCF?
Solution:
a and b are two prime numbers
Their HCF =1<\/p>\n
Question 31.
If a and b are relatively prime numbers, then what is their LCM?
Solution:
a and b are two prime numbers
Their LCM = a x b<\/p>\n
Question 32.
Two numbers have 12 as their HCF and 350 as their LCM (True \/ False).
Solution:
HCF of two numbers = 12
and LCM is 350
False, as the HCF of two numbers is a factor of their LCM and 12 is not a factor of 350.<\/p>\n
We have provided complete details of RD Sharma Class 10 Solutions Chapter 1 VSAQS. If you have any queries related to CBSE<\/strong><\/a> Class 10, feel free to ask us in the comment section below.<\/p>\n
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Class-10-Solutions-Chapter-1-VSAQS.jpg\")
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